Calc Review

https://www.coursera.org/learn/machine-learning-calculus/home/welcome

Derivatives

Motivation

  • Problem: Measuring velocity of car at given time
  • Solution: Estimate average velocity given 2 points (secant), average = rise / run
    • Secant: A line that intersects a curve at a minimum of two distinct points

Derivatives and Tangents

  • Solution to above problem: Move the distance between rise and run so close that you can't tell the distance between the two.
    • Derivate: Slope of the tangent line at a specific point in a function, $dx/dt$
    • Instantaneous rate of change: Slope of the tangent line

Slopes, maxima and minima

Zero Slope

Maxima / Minima

Takeyaways:

  • You can find the maxima / minima of a function by finding the zero slope (ie. tangent line is horizontal)

Derivatives and Notation

Lagrange vs Leibniz Notation

Some Common Derivatives

Constant (horizontal line)

Line (diagonal straight line)

Solution: $\frac{\Delta y}{\Delta x}$

Quadratics

Deriving the slope at (1, 1):

Formula:

Higher-degree polynomials

Cubed

Deriving the slope at (0.5, 0.2):

Formula:

Other Power Functions

Calculating the slope between (1,1) and (1.5, 2/3):

Deriving the slope at the limit of (1,1):

Formula:

Summary

Pattern for power functions:

  1. Exponent becomes the derivative as a multiplication factor
  2. Subtract one from the exponent and you get the new exponent

Common Derivatives

Inverse Function

What's an inverse?

  • Undoes the original function $f(x)$ by applying $g(x)$.
  • For $x$ becoming $x^2$, $g(x)$ applies a square root to $x^2$
  • Notation on the right (notice the $-1$ on $f$

Derivative of the Inverse

Takeaways:

  1. $f(x)$ and $g(x)$ mirror each other
  2. Their slopes, $\frac{\Delta f}{\Delta x}$ and $\frac{\Delta g}{\Delta y}$ essentially mirror each other
  3. Therefore, the derivative of $g$ ($g'(y)$) is the inverse of the derivative of $f$ ($f'(x)$): $$ g'(y) = \frac{1}{f'(x)} $$

Using an example at (1,1):

Using an example (2, 4):

Derivative of Trigonometric Functions

Sine

Cosine

Meaning of the Exponential $(e)$

$$ e = 2.71828182 $$

Convergence of $e$:

  • Another way to define $e$ is $(1 + \frac{1}{n})^n$
  • As $n$ reaches infinity, it becomes $e$
  • Derivative of $e$ is the same ($e^x$)

Intuition using bank interest as an example:

Derivative of $e^x$ is $e^x$

Proving derivative using secants:

Derivative of $log(x)$

Properties of logarithm

  • Logarithm of $x$ ($log(x)$) is the inverse function of $e^x$
  • The derivative of $f(x) = e^x$ as (2, 7.89) is $e^2$
  • The derivative of its inverse (7.89, 2), ie $log(y)$ is $\frac{1}{e^2}$
  • The derivative of $log(y)$ is $\frac{1}{y}$

Existence of the Derivative

Non-differentiable functions: Functions where you can't calculate a derivative at every point

  • For a function to be differentiable, the derivative has to exist for every point in the interval
  • Non-differentiable functions have a point where any tangent could work.
  • Examples: absolute ($|x|$), piecewise functions, functions with a vertical tangent ($f(x)=x^\frac{1}{3}$)

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